I have one ABM, 3 Schallers, 1 Mucho Grande brass, and several other high mass bridges. I have owned a BA on a Precision so I've experienced just about all you can with bridges.

IMO when you are talking about the high end aftermarket bridges, you are pretty much getting the same from each in regard to tone and sustain. I can't myself, hear any differenc in tonal coloration and I think that would apply to most bassists. The differences in material just don't seem to make an audible difference. I'm sure there is some differnence but it can only be measured by fine test equipment and therefore becomes pretty esoteric. What you WILL find with each type of bridge is a slightly different take on functionality. The ABM is a "quick release" type that allows easy string replacement. The ABM also uses a simple but very effective track system to corral the saddles in their fore-aft movement. The ABM has a very compact height - there aren't any components that stick up above the bridge to hang on things like case liners etc. ABM's also allow string spacing adjustments - an important feature for an aftermarket bridge since they would be used with different neck widths.

The BA's (I or II) probably weigh just a little more than the ABM and use a threaded screw for intonation adjustments. BA's are chrome plated brass. This doesn't have an effect on tone but is applied to keep the brass from oxidizing and getting cruddy-an easy thing with sweaty hands. Unlike the ABM the BA requires passing the string through a hole then over the saddle. This can sometimes be a bit of a problem with thicker gauge strings that need to bend in the backwrap portion just ahead of the brass stop. If your particular bass requires setting the saddle close to the back end of the bridge, you can sometimes run into some resistance pulling the saddle back. This isn't poor design and isn't the case with all string sets but it does happen with certain gauges especially on B and E strings. String spacing is done by locating each string on it's saddle and then filing a groove to set it in place. After that, the bridge will forever be setup for that instrument.

Then comes the Schallers. Though they look totally different from either the ABM or BA, they do the same thing with one important difference. Schallers let you adjust string spacing practically on the fly - something not available with the ABM's or BA's. This is done with a grooved ring threaded on a cross screw and captured in the saddle itself. By loosening the string then rolling the ring across the screw. Unlike the ABM's, Schallers aren't a quick-release bridge and require string through the back of the chassis like the BA's. Schallers also aren't aluminum or brass but rather a high density pot metal with a choice of finishes. Even this difference doesn't have an audible affect on tone IMO. I've also found that Schallers can be modified to allow stringing through the body but this doesn't apply to Fender instruments that already have through body stringing. You would have to establish your own hole placement to use this modification.

So, after all is said and done, IMO it's going to boil down to aesthetics, and cash. ABM's are fabulous looking but cost over $100 each. BA's look great but are over $50 and the Schallers look a bit busy but are usually under $50. I prefer Schallers for my instruments because of the great functionality at a reasonable price (for me) but your mileage will vary.